High Power CW Wattmeter Calibration at NIST

Transcription

1 [J. Res. Natl. Inst. Stand. Technol. 97, 673 (992)] High Power CW Wattmeter Calibration at NIST Volume 97 Number 6 November-December 992 Gregorio Rebuldela and Jeffrey A. Jargon National Institute of Standards and Technology, Boulder, CO The National Institute of Standards and Technology has established a measurement capability to support high power systems and devices. The automated wattmeter calibration system operates at power levels of to 000 W for frequencies from to 30 MHz and to 500 W from 30 to 400 MHz. A cascaded coupler technique is used to extend power measurements to high levels which are traceable to a 0 mw standard thermistor mount. This technique uses an arrangement of nominal 0, 20, 30, 40, and 50 db couplers with sidearm power meters. The initial step transfers the calibration of the 0 mw standard to the 0 db coupler/power meter. The standard is then replaced with a wattmeter to be calibrated. RF power is increased 0 db and the calibration is transferred to the adjacent 20 db coupler/power meter. This sequence is repeated with the remaining coupler/ power meters until the wattmeter is calibrated at the desired power levels and frequencies. Power ratios calculated from simultaneous power measurements made at each transfer are used to calculate the incident power at the wattmeter. Due to nonideal components, corrections are made for nonllnearities, mismatch, and other errors. Two types of wattmeters have been evaluated at selected frequencies and power levels. Total uncertainties are based on the random and systematic components. Key words: automated; calibration; cascaded; continuous wave; coupler; high power; measurement; transfer; uncertainty; wattmeter. Accepted: August 25, 992. Introduction There has been a recent interest in and demand for improved high power calibrations to support new and more accurate high power systems and devices being developed by industry. NIST has established a measurement capability to provide a traceability for continuous wave (cw) high power measurements. This paper describes the system, measurement scheme, calibration results and uncertainty analysis of the calibrations performed on different types of high power wattmeters. 2. System Description A diagram of the system is shown in Fig.. The rf source provides a stable rf signal at power levels of to 000 W for frequencies from 0 to 30 MHz, and to 500 W from 30 to 400 MHz []. The frequency and output power are controlled by software. A closed-loop feedback arrangement maintains the output power within ±0.005 db. The rf power path is switched to one of three test output ports depending on the frequency. Since the source delivers a minimum of W and the initial two calibration stages are made at 0 and 00 mw, an in-line attenuator is inserted between the source and the 0 db coupler to reduce the power to the required levels. This latching attenuator has a range of to 3 db in db steps and is controlled manually. 673

2 RF l=ill 0 db JC 20 db =)llf~30 db =lll 40 db ^C 50 db ^ TTTTT itt-_, i Coupler Coupler Coupler Coupler Coupler C J Group II : Watt- : meter 30 db ^ Display Group I Wattmeter Load Sensor Fig.. Block diagram of NIST high power cw wattmeter calibration system. The cascaded coupler arrangement is composed of nominal 0, 20, 30, 40, and 50 db directional couplers with sidearm power meters connected to digital voltmeters. Five coupler/power meters are required to transfer powers from 0 mw to 000 W in 0 db steps. Each sidearm power meter, composed of a tliermistor mount in conjuction with a NIST Type IV bridge, is connected to a digital voltmeter to measure rf powers within the bandwidth of the thermistor mount. The switcher connects each voltmeter to one of seven power meters depending on the stage of the calibration. A calibrated thermistor mount serves as the 0 mw reference for extending measurements to higher levels. Measurements are performed on two types of wattmeters. Group I includes three similar commercial units that measure rf power directly using diode power sensors. These sensors, used in conjunction with a power meter as a display, are microprocessor-based, each carrying its own wideband calibration constants in a self-contained nonvolatile memory. Since the calibration data are stored in the sensor, any sensor may be used with any power meter. Group II consists of two 30 db couplers each with a manually switched, 0-3 db step attenuator and thermistor mount on the sidearm. The attenuation is determined by the rf power incident on the coupler. The computer controls the rf source, the digital voltmeters, and the switcher, and handles the data acquisition and processing through an IEEE-488 bus. 3. Measurement Methods 3. Cascaded Coupler Technique At NIST, the measurement of rf power below GHz has been limited to 0 mw with thermistor mounts at uncertainties of ± 0.5% or better. A cascaded coupler technique, developed by K. E. Bramall [2], extends measurements to higher levels which are traceable to a 0 mw standard. Each stage is summarized below. Stage The 0 mw standard is connected to the cascaded coupler arrangement as shown in Fig. 2(a). Since the source delivers a minimum of W, an in-line attenuator is inserted between the source and the input of the 0 db coupler/thermistor mount combination to prevent any damage to the reference standard. With the attenuator set to 20 db, approximately 0 mw are applied to the reference standard. Simultaneous readings are taken on Ml and Ms. The power on Mi is nominally mw and due to the insertion loss of the coupler chain, Ms will indicate slightly less than 0 mw. 674

3 Power W MI M2 M3 Ui M5 0 db 20 db 30 db *0 db 50 db Coupler Coupler Coupler Coupler Coupler (a) Stage Ma Power ^ y 0 db W Ml M2 M3 M4 M5 ^ 0 db 20 db 30 db 40 db 50 db Coupler Coupler Coupler Coupler Coupler (b) Stage 2 Mx Poirer &- TV M2 M3 M4 M5 20 db 30 db 40 db 50 db Coupler Coupler Coupler Coupler (c) Stage 3 Mx Power & 0 w (d) Stage 4 M3 M4 M5 30 db 40 db 50 db Coupler Coupler Coupler Mx Power & 00 w (e) Stage 5 M4 40 db 50 db Coupler Coupler M5 J ; Mx Power & 500 vr (f) Stage 6 M5 50 db Coupler Mx Power & 000 w is) stage 7 M5 50 db Coupler Mx Fig. 2. Cascaded coupler arrangements for power transfer from 0 raw (o 000 W. 675

4 Stage 2 The 0 mw standard is replaced with the wattmeter, Mx, to be calibrated as shown in Fig. 2(b). If the wattmeter is from Group II, its attenuator is set to 0 db, and the rf power is increased 0 db to 00 mw, by setting the in-line attenuator to 0 db. Simultaneous readings result in a nominal 0 mw on Mj and mw on Mj. The main-arm output power, Pt, incident on the wattmeter is approximately 00 mw and is given by P ^KEl "- Px KB' () where Pi is the reading of the sidearm power meter of the 0 db coupler/thermistor mount. Mi, when the calibration was transferred from the 0 mw standard, Pi' is the reading of Mi when it was used to transfer the calibration to M2, and Ps is the reading of Ms from the first stage. The calibration factor, ^B, of the 0 mw standard is defined as the ratio of the substituted dc power in the thermistor mount to the cw rf power incident upon it. Equation () is true only if the impedances of the power standard and the wattmeter are equal. Since they are not, the expression is modified to include the effects of mismatch [3]. p ^t' -Ps i-roers ^ ^'- Pi /JTB li-rcerxp' (2) where fs and Fx are the reflection coefficients of the standard and wattmeter, respectively. The factor, TOE, is defined by Engen [4] as the equivalent generator reflection coefficient, and is given in terms of the coupler chain's scattering parameters, where ri ri AB MM = li-foerxp (4) (5) At this level, the calibration factor, Kn, of the Group I wattmeter is defined as Kn^ Px (6) where Px is the reading on the wattmeter's display. The calibration factor, Kn, of the Group II wattmeter is defined as Kf2 PL (7) where Px is the substituted dc power of the wattmeter's thermistor mount. In both cases, PL is the rf power incident on the wattmeters and is calculated at each rf power level. Stage 4 The 20 db coupler/thermistor is removed as shown in Fig. 2(d), and the source power is increased 0 db, to about 0 W. Simultaneous readings are taken on M3, M4, and Mx. The reading on M3, called P3', is about 0 mw and P^, the reading on M4, is about mw. The wattmeter is calibrated at this level using either Eq. (6) or (7), depending on the wattmeter. The main-arm output power, Pu is nominally 0 W and is given by Pl Pl Pi Ps PL= Pl Pl Py K. MM. (8) rce^sii SiiSji 53 ' (3) where the input of the 0 db coupler is port, the output of the 50 db coupler is port 2, and the sidearm of the 0 db coupler is port 3. Stage 3 The in-line attenuator and the 0 db coupler/thermistor is removed, as shown in Fig. 2(c). The calibration of the 20 db coupler/thermistor is not affected since a directional coupler has the property that the power split between the main and sidearm is independent of the source characteristics [5]. The source is set to W, a 0 db increase from the previous stage, and simultaneous readings are taken on M2, M3, and Mx. The reading on M2, referred to as Pl, is about 0 mw, while ft, the reading on M3, is approximately mw. The main-arm output power, Pu is about W and is given by 676 Stage 5 The 30 db coupler/thermistor is removed as shown in Fig. 2(e). If the wattmeter is from Group II, its attenuator is set to 0 db to prevent damage to its thermistor mount from subsequent increases of power. The source power is increased 0 db, to about 00 W and simultaneous readings are taken on M4, M5, and Mx. The reading on M4, called P/, is about 0 mw, and Ps is about mw. The main-arm output power, Pu is nominally 00 W and is given by a -P'' ^2 P3 P\ Ps,^,^ ^'^T; TIT;T, T.^^- (9) Stage 6 The 40 db coupler/thermistor is removed as shown in Fig. 2(f). If it is desired to calibrate the wattmeter between 00 and 000 W, such as 500 W, the rf power is increased by 7 db. If the wattmeter is from Group II, its attenuator is set to

5 7 db before applying rf power. Simultaneous readings are taken on Mj, called P^', and Mx. The main-arm output power is given by D -^l' -^2' -^3 Pi, P5' Ps,^,r /iri\ Stage 7 The source power is increased by 3 db, to 000 W, using the same configuration as the previous stage. See Fig. 2(g). If the wattmeter is from Group II, its attenuator is set to 20 db, and simultaneous readings are taken on M5, called P5", and Mx. The main-arm output power is given by The initial and final dc measurements are used with the Fon measurement to calculate the power and correct for any mount drift, which is assumed to be linear. The calculated value of Foff in Eq. (2) is given by Fo =Fo,f,i+ 3^ (Foff.f-Foff,i), (3) where Foff,i is the voltage reading taken before rf is applied at time ti, Vofu is the voltage taken after rf is removed at time ^3, and t2 is the time at which Fon is taken. '"-KKK^ftft""- <") The wattmeter is now calibrated at, 0, 00, 500, and 000 W at the desired frequency. 3.2 Modiflcations to the Cascaded Coupler Technique Since the high power source is limited to 500 W at MHz, wattmeters from Group I were calibrated at, 0, 00, 300 and 500 W in this frequency band. This still requires seven stages in the calibration although stages 6 and 7 are modified for lower powers. Wattmeters from Group II are rated at W, so they were calibrated at, 0, 00 and W which required six stages. When the measurements were taken, a 0 db coupler was not available, so a 4 db coupler was used instead. The only modification necessary was to set the in-line attenuator to 6 db rather than 0 db at the second stage, so enough power would be applied to the thermistors. 3.3 Power Measurements The NIST Type IV power meter does not directly read dc power in watts and must be connected to an external dc voltmeter.the substituted dc power, Pic, is calculated from measured voltages and is given by Pdc (2) where Vou is the output voltage with no rf power applied, Fon is the output voltage with rf power applied, and /?o is the operating resistance of the thermistor mount. Figure 3 shows the measurement sequence for a power calculation [6]. An initial Vou is taken; rf power is then applied and Fo is measured; rf power is removed and a final Foff is taken. 677 Fig. 3. Sequence for measuring power meter dc voltages. 4. Measurement Results Measurements were made on both groups of wattmeters at several frequencies and power levels. Group I wattmeters were calibrated at, 0, 00, 500 and 000 W at frequencies from 2 to 30 MHz and at, 0, 00, 300 and 500 W at frequencies from 30 to 400 MHz. Group II wattmeters were calibrated at, 0, 00 and W at the same frequencies. The calibration factors for a Group I wattmeter are'near unity at all power levels since it measures power directly with a diode detector. A Group I wattmeter has one sensor, denoted Sensor, that measures powers at frequencies between.8 and 32 MHz and another. Sensor 2, that measures power at frequencies between 25 and 000 MHz. Sensor was used at frequencies between 2 and 30 MHz, and Sensor 2 was used at frequencies between 35 and 400 MHz. Table lists calibration factors at selected frequencies for three wattmeters from Group I. The calibration factors differ among wattmeters, and the calibration factor at each frequency increases with power, partly due to nonlinearity in the diode detector.

6 The calibration factors for a Group II wattmeter range from,000 to 20,000 due to the 30 db directional coupler and the attenuator's setting which is dependent on the power level; 0 db at and 0 W, 0 db at 00 W, and 3 db at W. One wattmeter has a frequency range from 2 to 00 MHz and the other has a range from 00 to 400 MHz. Tables 2 and 3 list the measured calibration factors of the two wattmeters. Table. Calibration factors of Group I wattmeters Freq. Power level Wattmeter A Wattmeter B Wattmeter C (MHz) (W) cal. factor cal. factor cal. factor Table 2. Calibration factors of Group H-A wattmeter Freq. (MHz) Power level (W) Table 3. Calibration factors of Group II-B wattmeter Freq. (MHz) Power level (W) Wattmeter A cal. factor Wattmeter B cal. factor

7 5. Uncertainty Analysis 5. Systematic Uncertainty The factors contributing to tlie total systematic uncertainty are: a. Uncertainty in the dc voltmeter measurements. b. Uncertainty in the Type IV power meters. c. The dual-element substitution errors associated with the coaxial thermistor mounts. d. Uncertainty in the 0 mw standard mount calibration factor. e. Mismatch uncertainty due to the reflection coefficient of the 0 mw standard mount, the reflection coefficient of the wattmeter/high power load combination, and the equivalent generator reflection coefficient. f. Nonlinearities in the cascaded couplers. g. Uncertainty in the high power source. 5.. Voltmeter Uncertainty The uncertainty in the individual voltmeter readings can be determined by taking the total differential of the power expression, Eq. (2), which gives dp=^ (KoffdFoff-FondFon), (4) The total differential of power, Eq. (4), can be determined by taking the differential of Voa, Eq. (3), which gives where dfoff=(l-t)dfofl.,- + TdFoff,/, r= t2-tl t3-tl (5) (6) The uncertainties, dfoff.i, dfoff,f, and dfon, in the measured values of Fofr.i, Foff.r and Voa, are based on the voltmeter manufacturer's specifications. Figure 4 shows the uncertainty in the power measurement as a function of power level, assuming the coupler sidearm powers, Pi through Ps and Pi' through Ps', are ratioed as in the Bramall measurements. Figure 5 shows the uncertainty when a power is not ratioed as in the case of Ps. The power measurements, Pi' through Ps', are approximately 0 mw, which result in uncertainties of 0.0%. The power measurements, Pz through Ps, are approximately mw, which result in uncertainties of 0.07%. The measurement of Pi is about 0.4 mw due to the 4 db coupler and has an uncertainty of 0.7% Type IV Power Meter Uncertainty The four possible sources of uncertainties internal to the Type IV power meter are the reference resistors, the operational amplifier open-loop gain, input offset voltage, and input bias current. Larsen has shown that the uncertainties due to the Type IV power meters are negligible compared to those of the voltmeters [7] Dual-Element Uncertainty The thermistors used in the system are dual-element bolometers. They are nonlinear with power due to the rf-dc substitution error that occurs because the two elements are not identical [8]. The NIST calibration of the effective efficiency is done at 0 mw; therefore, this error is of concern when measurements are made below this power. Direct measurements were performed on similar thermistor mounts [6] resulting in a nonlinearity of about 0.04% at the mw level Uncertainty in the Standard Mount Calibration Factor The uncertainty of the NIST thermistor mount calibration factor, KB, is approximately ±0.5% in the worst case. The 0 mw standard is recalibrated periodically Mismatch Uncertainty Since the impedances of the standard power meter and the high power load are not equal, mismatch is introduced when the power meter is replaced by the load. The mismatch term, discussed earlier, is given by (7) The uncertainty of the mismatch term requires the knowledge of the uncertainties in measuring Fx, -Ts, and the couplers' scattering coefficients. These uncertainties are given in Table 4. The uncertainty of FOE, which is almost entirely due to the uncertainty of 52, is ± and is combined with those of the 0 mw standard and wattmeter/load combination to calculate the total mismatch uncertainty. Two different methods were used to analyze the uncertainty. First, a simulation program was written to calculate the mismatch uncertainty using random values of magnitude and phase, within their respective limits, for the reflection coefficients along with their respective uncertainties. Several hundred trials were performed, resulting in a maximum mismatch uncertainty of ±0.9%. Second, the mismatch uncertainty was calculated by combining the terms in Eq. (7) in the worst phase with the uncertainties included. The result 679

8 was a maximum mismatch uncertainty of ±0.2%. The latter method was arbitrarily chosen and its derivation is explained in the Appendix Nonlinearity of Couplers The directional couplers were chosen with power ratings greater than the actual requirements to minimize the power sensitivity of the couplers. Each coupler is rated at least one and one-half times its maximum applied power. Tests for power nonlinearities were performed on selected couplers at higher powers, and an estimate for the entire coupler chain is approximately ±0.30% Uncertainty in the High Power There are several uncertainties due to the radio frequency source, most of which are negligible. a. Harmonics are at least 46 db below the fundamental signal at the output port, thus having negligible effects. b. Spurious signals are also negligible since they are approximately - 60 dbc. c. The frequency uncertainty is approximately ±0.00% due to the internal free-air crystal oscillator of the rf source. d. The rf source amplitude stability is specified by the manufacturer to be ±0.2%. X O i. LJ PoLue r, m i i Luatts Fig. 4. Power measurement uncertainty due to DVM when ratios are taken. 680

9 35 c\* Ti-nrrrTTtTrTTtTTi-i PoLue r, m i Luat ts 0 Fig. 5. Power measurement uncertainty due to DVM when ratios are not taken. Table 4. Reflection coefficients and uncertainties of mismatcii components Reflection coefficient of 0 mw std. Reflection coefficient of wattmeter/load combination Reflection coefficient of equivalent generator 522 of coupler chain 52 of coupler chain 532 of coupler chain 53 of coupler chain Max. value ± uncertainty 0.02 ± ± ± db db db 68

10 5..8 Overall Systematic Uncertainty A summary of all the systematic uncertainty components and the total as calculated by the root-sum-square method are shown in Table 5. The overall systematic uncertainty is ±0.67%. Table 5. Systematic uncertainty components Uncertainty source Contribution {%) dc voltage measurements Measurement of Pi ±0.7 Measurement of P2 ±0.07 Measurement of Pj ±0.07 Measurement of P4 ±0.07 Measurement of Ps ±a.d7 Measurement of Pi' ±0.0 Measurement of P2 ±0.0 Measurement of P3' ±0.0 Measurement of P^' ±0.0 Measurement of Ps' ±0.0 Measurement of Ps ±0.03 Dual element of bolometer mounts Measurement of P] ±0.05 Measurement DfP2 ±0.04 Measurement of Pi ±0.04 Measurement of P4 ±0.04 Measurement of Ps ±0.04 Power standard calibration factor ±0.50 Mismatch due to reflection coefficients ±0.20 Nonlinearity of cascaded couplers ±0.30 High power source ±0.2. Total (RSS) ± Random Uncertainty Each of the wattmeters was calibrated five times to determine the repeatability of the measurements. Tests were made at various times of the day over several days to cover as many random factors as possible, including variations of environmental conditions and quality of the connections by the operator. The sample standard deviations were calculated for each meter at all frequencies and power levels. Table 6 lists the standard deviations of the three Group I wattmeters; Table 7 lists the standard deviations of the Group II-A wattmeter (2-30 MHz); and Table 8 lists the standard deviations of the Group II-B wattmeter ( MHz). Wattmeter C of Group I was calibrated five more times over a 6 month period to determine the longterm stability of the calibration factors. Figures 6, 7, and 8 show the ten measurements at each power level with their averages at 2, 00, and 400 MHz, respectively. Sample standard deviations of the ten trials ranged from 0.07% to 0.66%. 5.3 Total Uncertainty The total uncertainty, Ur, may be calculated by combining the standard deviation, S, determined from N repeated measurements, with the overall systematic uncertainty, A, using the equation (8) Table 9 lists the systematic uncertainty, ranges of values for the random uncertainties, and total uncertainties for each wattmeter. 6. Conclusion The calibration of high power cw wattmeters is accomplished using the cascaded coupler technique. Directional couplers are used to extend the range of low power meters up to the kilowatt range. Although this technique is quite cumbersome and lengthy due to multiple power transfers, the standard deviations are less than 0.66% over a 6 month period for Wattmeter C in Group I. Standard deviations for all other wattmeters vary from 0.03% to 0.80% and are caused largely by the instability of the individual wattmeter. The overall uncertainty limits are 0.77% to.05% depending on the type of wattmeter, frequency, and power level. Wattmeters may be used to calibrate a high power source for certifying other wattmeters, thus avoiding the cascaded coupler arrangement and reducing measurement time. However, this introduces another level in the calibration structure, resulting in higher uncertainties. 682

11 Table 6. Sample standard deviations of Group I wattmeters Table 7- Sample standard deviations of Group H-A wattmeter Freq. Power level Wattmeter A Wattmeter B Wattmeter C (MHz) (W) std. dev. % std. dev. % std. dev. % Freq. (MHz) Power level (W) Wattmeter A std. dev. % , , Table 8. Freq. (MHz) Sample standard deviations of Group II-B wattmeter Power level (W) Wattmeter B std. dev. %

12 ="=''v-::::. ^i---. "--t^ V - -A '*'--. ^ "'" V ' ''^-....-V - -V -- V--""" ' ---A - -A-'".^--' V 2.05 o CO ^.0 o CO -i [3-._ " 'O - -R' _...-C] "-D " 0-.. < >.-. "n--..--u--- D- --a 4 (>' ;-;:o-- '""-0-., -A '- <> ' Q watt ^0 watts 00 watts ^-500 watts ^ 000 watts o-' :-Ov ^, O ' j~\ --0 ""' o-'" ' " :-6 l,..l.. n...., >,, \ <,,, Trials Fig. 6. Calculated values of calibration factors (ten trials) for Group I-C wattmeter at 2 MHz and at various power levels. Averages of ten trials shown as solid lines. 684

13 o.05 LL O.0 *-> CO v_ <* i -005 (0 O -.-i?'' _,.v- "- f--' " i:: A ----A.., " ""-A' n-... ' A,-.^ """^_ "'A D-'.--D"'" :::-n-:;::; "" a-- " U--- rh --- -& --., CD- _...,^, ,,,^--. " --<> "- -' '\ V ""^--'-'-'.L i I'l-A,. o watt 00 watts c 00 watts A 300 watts V 500 watts o,. ^' n 0.99,, i,,, <,,,,,, Trials Fig. 7, Calculated values of calibration factors (ten trials) for Group I-C wattmeter at 00 MHz and at various power levels. Averages of ten trials shown as solid lines. 685

14 I.05 i ^ o CO ^ ^;^ V C-, - ^^,,.v,,.-' V-"" '.-A- ^Vr-''-.-A-.. -ti-- - /i V-.. -A []...-D u ".':- t(-''' - <r^ ~ -> 0 ^ y -^ ""V--, "-V '" """'--V - ~A' I::Cb-.=.=.g i "-0-,:-- --l-ix..-..,. -«--*^ -- o watt 00 watts 00 watts A 300 watts -V-500 watts 0.97 (>- n - O- r^--'.:-o.. ''~o-.. ' I.L.L..I..! Trials,, r;?:: Fig. 8. Calculated values of calibration factors (ten trials) for Group I-C wattmeter at 400 MHz and at various power levels. Averages of ten trials shown as solid lines. { Table 9. Systematic uncertainties and ranges of values for the random and total uncertainties of the wattmeters Group I Systematic Random Total uncertainty uncertainty uncertainty (%) (%) (%) Wattmeter A Wattmeter B Wattmeter C Group II Wattmeter A (2-30 MHz) Wattmeter B ( MHz) 7. Appendix A Since the impedances of the 0 mw standard and the wattmeter/load are not equal, a mismatch term, MM, is introduced [3] and is given by (9) where Fx and Ts are the reflection coefficients of the wattmeter/load combination and the power standard, respectively, and /CE is the equivalent generator reflection coefficient. The reflection coefficients are complex numbers and can be written in the form TGE = I TOEI (COS0GE +; sin^ge), (20) 686

15 rs = rs (cos&s+;sin0s), (2) 8. References rx= rx (cosftc+jsinflx) (22) where GOE, ^S, and dx are the arguments of the reflection coefficients of the equivalent generator, the power standard, and the wattmeter/load combination. The mismatch term is simphfied and approximated using several steps. First, completing the squares of both the numerator and denominator of Eq. (9) gives i-2rcers+(rcers)^ i-2rgerx+(rgerx)' (23) An approximation may be used by deleting the (JCEA)^ and (ros/s)^ terms since their contributions are negligible. This gives MM- l-2rgers GEJ S ' i-2rcerx (24) Expanding and neglecting the higher order terras, Eq. (24) can be written as l-2irge rs cos(flgb+fe) i_2jrge!rxlcos(0ce+ex) (25) The cosine terms can range in value from - to -f-. Therefore MM has a range l±2liodm ^^ i±2 rge rx - With the uncertainties included (26) ^^ i±2( rgel±4 rcel)(irv ±4 rxl)- ^^'^ Acknowledgments The authors extend their thanks to Robert Judish for discussions on uncertainty analysis. Manly Weidman and John Juroshek for assistance with the mismatch uncertainty calculations, and Neil Larsen and Fred Clague for supplying the graphs found in Figs. 4 and 5. [] Installation, Operation and Maintenance Instructions With Illustrated Parts List for Automated Wattmeter Calibration System, M/A-COM Microwave Power Devices, Inc., May 990. [2] K. E. Bramall, Accurate Microwave High Power Measurements Using a Cascaded Coupler Method, J. Res. Natl. Bur. Stand. (U.S.), 7SC (3-4), (97). [3] G. F. Engen, Recent Developments in the Field of Microwave Power Measurements at the National Bureau of Standards (U.S.), IRE Transactions on Instrumentation, -7, (958). [4] G. F. Engen, Amplitude Stabilization of a Microwave Signal, IRE Transactions on Microwave Theory and Techniques, MTT-6, (958). [5] R. W. Beatty and A. C. Macpherson, Mismatch Errors in Microwave Power Measurements, Proc. IRE 4 (9), 2-9 (953). [6] F. R. Clague, Power Measurement System for mw at GHz, Natl. Inst. Stand. Technol., Technical Note 345 (990). [7] N. T. Larsen, A New Self-Balancing DC-Substitution RF Power Meter, IEEE Trans. Instrum. Meas. IM-25, (976). [8]. G. F. Engen, A DC-RF Substitution Error in Dual- Element Bolometer Mounts, IEEE Trans. Instrum. Mcas. IM-3, (964). About the Authors: Gregorio Rebuldela has worked on the development and evaluation of high frequency, coaxial voltage standards and measurement systems and on the development of the low frequency dual six-port automatic network analyzer which measures the circuit parameters of one and two port devices. His current technical responsibilities as a project leader in the Electromagnetic Fields Divison at NIST include developing and evaluating cw, coaxial high power measurement systems and transfer standards, and improving and maintaining the high frequency rf voltage and low power calibration services. Jeffrey A. Jargon is a member of the Microwave Metrology Group in the NIST Electromagnetic Fields Division, where his main responsibilities are in cw, coaxial high power and high frequency voltage metrology. The National Institute of Standards and Technology is an agency of the Technology Administration, U.S. Department of Commerce. 687